By [1]}, an entanglement witness \(W\)  for an infinite
dimensional system is optimal if and only if \(W-D\)  can not be an
entanglement witness anymore for any nonzero operator \(D\ge 0\) . So,
if \(\hat{W}_{\hat{\Phi }^{(n,k)}}\)  is not optimal, then there exists
a nonzero positive operator \(\hat{D}\in {\mathcal {B}}(H\otimes K)\) 
such that \(\hat{W}_{\hat{\Phi }^{(n,k)}}-\hat{D}\)  is an entanglement
witness. Note that \((P\times Q)\hat{W}_{\hat{\Phi }^{(n,k)}}(P\otimes Q)=\hat{W}_{\hat{\Phi }^{(n,k)}}\) . Then
\(\hat{W}_{\hat{\Phi }^{(n,k)}}-\hat{D}\)  is an entanglement witness
implies that, for any separable pure state \(\sigma _{lh}=|l\rangle \langle l|\otimes |h^{\prime }\rangle \langle h^{\prime }|\)  with
\(l,h>n\) , we have
\(-{\rm Tr}(\hat{D}\sigma _{lh})={\rm Tr}((\hat{W}_{\hat{\Phi }^{(n,k)}}-\hat{D})\sigma _{lh})\ge 0,\) 
